A001725 a(n) = n!/5!. (Formerly M4243 N1772)

1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840, 726485760, 10897286400, 174356582400, 2964061900800, 53353114214400, 1013709170073600, 20274183401472000, 425757851430912000, 9366672731480064000, 215433472824041472000, 5170403347776995328000

Offset

5,2

Comments

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=6) ~ exp(-x)/x*(1 - 6/x + 42/x^2 - 336/x^3 + 3024/x^4 - 30240/x^5 + 332640/x^6 - 3991680/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 5..300

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 265.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.

D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

Index entries for sequences related to factorial numbers.

Index to divisibility sequences.

Formula

E.g.f. if offset 0: 1/(1-x)^6.

a(n) = A173333(n,5). - Reinhard Zumkeller, Feb 19 2010

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+6)/(x*(k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

G.f.: W(0)/(40*x^2) -1/(20*x^2) -1/(5*x) , where W(k) = 1 + 1/( 1 - x*(k+4)/( x*(k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013

a(n) = A245334(n,n-5) / 6. - Reinhard Zumkeller, Aug 31 2014

E.g.f.: x^5 / (5! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=5} 1/a(n) = 120*e - 325.

Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)

Mathematica

lst={}; Do[AppendTo[lst, n!/5! ], {n, 5, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
Range[5, 30]!/120 (* Harvey P. Dale, Dec 20 2014 *)

Prog

(PARI) a(n)=n!/120 \\ Charles R Greathouse IV, Jul 19 2011
(Magma) [Factorial(n)/120: n in [5..25]]; // Vincenzo Librandi, Jul 20 2011
(Haskell)
a001725 = (flip div 120) . a000142 -- Reinhard Zumkeller, Aug 31 2014

Crossrefs

a(n)= A049374(n-4), n >= 1 (first column of triangle). Cf. A049460, A051339. a(n)= A051338(n-5, 0)*(-1)^(n-1) (first unsigned column of triangle).

Cf. A245334, A000142.

Sequence in context: A367241 A262671 A029588 * A361547 A123510 A265871

Adjacent sequences: A001722 A001723 A001724 * A001726 A001727 A001728

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Dec 20 2014

Status

approved